Generic divisibility test (how-to)

A Vedic mathematics technique to test whether a number is divisible by N ( for N ending in 1, 3, 7, or 9 ).

The first step is to discover the osculator for N, by the vedic sutraएकाधिकेन पूर्वेन meaning By one more than the previous. To do this, find a number that, when multiplied by N, will yield a number Y ending with the digit 9. This is 9, 3, 7 and 1 for N ending with digits 1, 3, 7 and 9 respectively. Then, take the previous i.e. the part of Y excluding the last digit (9) and take One more than it i.e.increment it. Call this number O (the osculator).

N = 23 -> Y = 69 -> O = 6 + 1 = 7

N = 37 -> Y = 259 -> O = 25 + 1 = 26

N = 31 -> Y = 279 -> O = 27 + 1 = 28

Assuming the number we want to test is X, we apply the following process -

Chop the last digit off X (call this digit D) , multiply D by O and add to X.

If X = N then we are done and N divides X.

If X < 10N, Divide X by N and get the answer ( X will inevitably go below 10N eventually )

Repeat

Examples :

Is 175121 divisible by 37?

For 37 the osculator is 26.

17512 + (1 * 26) = 17538

1753 + (8 * 26) = 1961

196 + (1 * 26) = 222

22 + (2 * 26) = 74

Here we can stop since 74 is divisible by 37

Is 13174584 divisible by 23?

23's osculator is 7.

1317458 + 28 = 1317486

131748 + 42 = 131790 ( we can drop the zero here, since it will make no difference )

1317 + 63 = 1380 ( once again drop the zero )

13 + 56 = 69

69 is a multiple of 23 so we're done

This technique may not be the fastest possible, but it works for all numbers and can possibly be done mentally even for larger dividends. This same technique simplifies to addition of digits for 9 and 3, which both have an osculator of 1.

Vedic mathematics was discovered by Sri Bharati Krishna Tirtha (1884-1960), who took 16 cryptic sutras from the Vedas and elaborated them into mathematical techniques (he claimed revelation as a source). Many of the techniques (with practice) allow rapid mental aritmetic. He wrote sixteen volumes covering everything from arithmetic to more advanced algebra, differential/integral calculus, plane/solid geometry, hyperbolic functions and many other branches, but unfortunately the manuscripts were destroyed in a fire. Thereupon he started rewriting them from memory, but he passed away before he could write down anything more than an introductory volume. That volume is all that survives today and is published as a book.